Infinite Dimensional Geometric Singular Perturbation Theory for the Maxwell--Bloch Equations

Infinite Dimensional Geometric Singular Perturbation Theory for the Maxwell-Bloch Equations

We study the Maxwell–Bloch equations governing a two-level laser in a ring cavity. For Class A lasers, these equations have two widely separated time scales and form a singularly perturbed, semilinear hyperbolic system with two distinct characteristics. We extend Fenichel’s geometric singular perturbation theory [N. Fenichel, J. Differential Equations, 31 (1979), pp. 53–98] to the Maxwell–Bloch...

Geometric Singular Perturbation Theory

Geometric singular perturbation theory for stochastic differential equations

We consider slow–fast systems of differential equations, in which both the slow and fast variables are perturbed by additive noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the re...

Geometric singular perturbation theory in biological practice.

Geometric singular perturbation theory is a useful tool in the analysis of problems with a clear separation in time scales. It uses invariant manifolds in phase space in order to understand the global structure of the phase space or to construct orbits with desired properties. This paper explains and explores geometric singular perturbation theory and its use in (biological) practice. The three...

Geometric Singular Perturbation Theory for Non-smooth Dynamical Systems

In this article we deal with singularly perturbed Filippov systems Zε: